Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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# The Function F ( X ) = Sin ( π [ X − π ] ) 4 + [ X ] 2 , Where [⋅] Denotes the Greatest Integer Function, is (A) Continuous as Well as Differentiable for All X ∈ R - Mathematics

#### Question

The function $f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}$ , where [⋅] denotes the greatest integer function, is

##### Options
• continuous as well as differentiable for all x ∈ R

• continuous for all x but not differentiable at some x

• differentiable for all x but not continuous at some x.

• none of these

#### Solution

(a) continuous as well as differentiable for all x ∈ R

Here,

$f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}$

Since, we know that

$\pi\left[ \left( x - \pi \right) \right] = n\pi$
$\ \text { sin n} \pi = 0$
$4 + \left[ x \right]^2 \neq 0$
∴f(x) = 0 for all x

Thus, f(x) is a constant function and it is continuous and differentiable everywhere.

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